Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear...
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![Page 1: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/1.jpg)
Elementary Operations of MatrixElementary operations of matrix are an important tools in linear algebra.
The next three operations are called as No.1st,2ndand3rd elementary transforms of matrix.
)(or )( :follows as
and ns)rows(colum einterchang we)(
jijiijij ccrrcrj
ii
).(:follows as
kby )row(column hemultiply t we)(
ii kckr
iii
).(:)asrow(columnith the
to)row(columnjth thek times add we)(
jiji kcckrr
iii
![Page 2: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/2.jpg)
4131
122122
2832
A
2832
122122
4131
31 rr
6690
4460
41311312
22
rrrr
2230
2230
4131
0000
2230
4131
.BAor BA asmark we
operations elemantaryby
B intoA ormcan transf weIf
,
How to transform a matrix into an echelon oneFunction
Elementary operations of matrix about row and column are all called elementary operations
.
![Page 3: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/3.jpg)
301
020
201
A
500
020
201
9113
12334
3221
B
0770
011110
3221
0000
0110
3221
100
010
001
0000
0010
0001
![Page 4: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/4.jpg)
Equivalent Matrix If matrix A can be transformed into B by some elementary operations, we say A and B are equivalent,and we marked as A B.
Equivalent Matrices have the following characters:CACBBAABBAAA ,;;
nmIA
00000
00000
00100
00010
00001
normal form of A
Theory: any matrix has its own
normal form.
![Page 5: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/5.jpg)
As last e.g.:
0000
2230
4131
A
0000
2230
0001
1413
12
4
3
cccc
cc
0000
0010
0001
32
32
3
2
24
23
r
cc
cc
Corollary:matrices A and B are equivalent if and only if they have the same normal forms.
![Page 6: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/6.jpg)
Rank of Matrix
A. ofnant subdetermik order an
called is scalars ofarray resulting theout, picked are
columnsk and rowsk matrix.Ifn man beA Let .1
.in nantssubdetermiorder - are There nmkn
km AkCC
2.The highest order of nonzero subdeterminantsof A is called the rank of A and denoted by r(A).
Obviously:r(O)=0.As long as A is not 0, r(A)>0. And :};,min{)()( nmAri nm
.)(
zero, are nantssubdetermiorder -r all if;)(
nant,subdetermiorder -r nonzero a is thereIf)(
rAr
rAr
ii
![Page 7: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/7.jpg)
).()()( ArAriii T e.g.: Compute the rank of Matrix A.
00
00
2222
111211
rnrr
nr
nr
aa
aaa
aaaa
A
.)(Obviously rAr
We can figure out the rank of the matrix by elementary operations
)0( 2211 rraaa
![Page 8: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/8.jpg)
The Rank:Theory: Elementary operations do not change the resulting scalar of the rank.
Prove: only after elementary row operations.
);()( BrArBAijr
);()( BrArBAikr
mnmm
jnjj
inii
n
aaa
aaa
aaa
aaa
A
21
21
21
11211
ji krr
![Page 9: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/9.jpg)
B
aaa
aaa
kaakaakaa
aaa
mnmm
jnjj
jninjiji
n
21
21
2211
11211
We can conclude:
)()()()( BrArBrAr )()( BrAr
![Page 10: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/10.jpg)
e.g. To figure out the rank of matrices
4131
122122
2832
.1 A
2832
122122
4131
31 rrA
6690
4460
4131
0000
4460
41312)( Ar
![Page 11: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/11.jpg)
9300
121070
2220
4321
9300
5300
1110
4321
9300
121070
1110
4321
4000
5300
1110
4321
4)( Br
5021
0113
2101
4321
.2 B
![Page 12: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/12.jpg)
9113
1234
3221
tA ?, 3)(? ifonly Art
0770
01180
3221
tA
01180
0110
3221
t
0300
0110
3221
t
.3)(,3 Art
![Page 13: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/13.jpg)
Elementary Matrices
Definition: An elementary matrix is one obtained by performing a single elementary operationon an identity matrix.The followings are three different kinds of elementary matrices:
1
01
10
1
),(
jiE
![Page 14: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/14.jpg)
They also include the ones obtained by performing a single elementary column operation on an identity matrix.
1
1
))((
k
kiE
1
1
1
1
))(,(
k
kjiE
![Page 15: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/15.jpg)
Properties of Elementary Matrices1.
),(),( jiEjiET
1
01
10
1
),(
TjiE ),( jiE
![Page 16: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/16.jpg)
))((
1
1
))((
kiE
kkiE T
))(,(
1
1
1
1
))(,( kijE
k
kjiE T
))(())(( kiEkiET
))(,())(,( kijEkjiET
![Page 17: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/17.jpg)
Transposes of elementary matrices are sameto themselves.
1),( jiE2. kkiE ))(( 1))(,( kjiE
Elementary matrices are all invertible.
The Relation Between Elementary Matrices and Elementary Operations.
Look at an example first.
![Page 18: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/18.jpg)
333231
232221
131211
aaa
aaa
aaa
A
100
001
010
)2,1(E
333231
232221
131211
100
001
010
)2,1(
aaa
aaa
aaa
AE
333231
131211
232221
aaa
aaa
aaa
![Page 19: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/19.jpg)
Elementary row operations are equivalent to multiply anelementary matrix on the left. And multiplying an elementary matrix on the right stands for Elementary column operations.
nmmmrownmrow AETAT )()(
)()( nncolumnnmnmcolumn ETAAT
100
001
010
)2,1(
333231
232221
131211
aaa
aaa
aaa
AE
333132
232122
131112
aaa
aaa
aaa
![Page 20: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/20.jpg)
e.g. To find the normal form of matrix A and useelementary matrices to show the elementary operations
011
110
001
A
010
110
001
13 rr
110
010
001
32 rr
100
010
001
23 rrI
101
010
001
1P
010
100
001
2P
110
010
001
3PIAPPP 123
We can prove it
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333231
232221
131211
aaa
aaa
aaa
A
133312321131
131211
232221
aaaaaa
aaa
aaa
B
100
001
010
1P
101
010
001
2P
BPAP 21)1(BPAP 12)2(
BAPP 21)3(BAPP 12)4(
![Page 22: Elementary Operations of Matrix Elementary operations of matrix are an important tools in linear algebra. The next three operations are called as No.1.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e375503460f94b26e79/html5/thumbnails/22.jpg)
4131
122122
2832
101
010
001
1000
0120
0010
0001
6903
122122
2832
1000
0120
0010
0001
69183
122162
28192